ntent [at. %]Zn con

diffusion process at migrating interface of discontinuous precipitates

of discontinuous precipitates

1. Introduction

2. Global characterization of the DP reaction 3. Unsolved problems

4. Local characterization of the DP reaction via AEM

5. Principles of high resolution microchemical analysis of lamellar structures

6. Grain boundary diffusivities via AEM

4. Local characterization of the DP reaction via AEM

7. Concluding remarks

Discontinuous Precipitation (DP) Reaction

L

T

α β

v _RF

δ

λ_α

The solute redistribution occurs at the moving RF.

α+β α β

x_o

A ^xexav x_B B

GB

α

α+β

1. There exists an excess of solute atoms within the α lamella compared to the equilibrium state

Global Concept of DP Reaction

All the parameters represent the average values for the whole population of the cells in the sample

Quantitative metallography: λλλλ_αααα,,,, v_av

X-ray diffraction: average solute concentration in the α lamellae: x_av

w

Time

v_av

w w=πw/4

Cahn´s model

Diffusion Models - Global Concept

tanh 2

2 C

x C x

x x

e o

av

o =

−

−

C D v

s _b ^av

α2

δ

λ

Petermann- Hornbogen (P-H) model

Z. Metallkde 59 (1968) 814 G

RT D v

s _b ^av

= ∆ 8

2

λα

δ

D_b-grain boundary (GB) chemical diffusion coefficient s-segregation factor

∆G-driving force for the DP reaction: ∆G = f(x_av, x_o, x_e) R-gas constant

T-absolute temperature of the DP reaction

x_o,x_e-solute content in the alloy and at the α/β interface

Questions

Why?

(Petermann-Hornbogen model) (Cahn´s model) sδD_b ≈ 10²...10⁴ x sδD_b

Does diffusivity at the migrating grain boundary (GB) occur at the same speed as for the stationary GB?

Local Concept of DP Reaction

All the parameters are relevant for the individual set of α α α α and β β β lamellae (single cell) β

x(y)

λ λ λ λ_αααα

y

0 1

DP - Solute concentration profile across the αααα lamella

λ λ λ λ_αααα

v_ins

RF

( )

[ ]

o o

e x

C

C x y

x y

x − +

−

= cosh( /2)

5 . 0 ) cosh

( ) (

b ins

D s C v

δ λ

=

o b

x s = x

J.W. Cahn: Acta Metallurgica 7 (1959) 18

[ ( ) ] ⁰

2 2

=

−

− _o

b

b v x y x

dx x D d

δ Boundary conditions: For y=0 and y=1 y(x) = x_e

x(y)

λ λλ λ_αααα

y

0 1

C /2)

cosh( _o

v_ins – instantaneous growth rate, λ

λλ

λ_αααα – thickness of αααα lamella,

x_e – equilibrium concentration at α/βα/βα/βα/β interface, x_o – solute content in alloy,

δδδ

δ – width of grain boundary, s - segregation factor,

sδδδδD_b – diffusivity at moving grain boundary

Growth velocity of the DP reaction

0 s ^RF 4 s

In-situ observation in TEM

w

v_av v_ins

Al-22 at.% Zn aged at 450 K 12 s

0.2 µm 15 s

Stop- and –Go
vav

only Go



vins

v_ins >>v_av

Czas

Two-step ageing procedure

T₂ >T₁ w T₁

725K

P. Zięba, W. Gust, Acta mater. 47, (1999) 2641 Ni-4 at.% Sn alloy aged for 250 h at 725 K

followed by 60 h at 775 K.

775 K

 The α lamellae exhibited approximately the same thickness for a sufficiently long distance;

 The thickness of several neighbouring α lamellae within the same colony remained more or less the same;

 No distinct changes of the reaction front velocity within the same colony

within the same colony

 Any case of increasing the α lamella spacing, preceding branching or re-nucleation of the new β lamella, is not taken into account;

 The influence of the stop- and -go fashion of the

reaction front movement on the solute

concentration profiles is avoided.

L

Al-Zn system

α

β

x_o

4 6 8 10

α α α α

RF

Al-22 At.% Zn

0 40 80 120 160

2 4

Spacing [nm]

0.2 µm

EDX analysis after DP reaction in the temperature range 350-475 K and for 5 different colonies.

Region of cells

Lamella analysed

λ_α (nm)

C x_i

(at.% Zn)

y(x=0.5) (at.% Zn)

v (nm/s)

sδD_b (m³/s)

1 1

2 3 4 5

210 225 205 195 240

2.14 1.66 2.51 1.95 2.21

3.28 4.32 3.74 3.51 4.07

7.38 7.44 8.27 7.29 8.08

86 1.8 × 10^-21 2.6 × 10^-21 1.4 × 10^-21 1.7 × 10^-21 2.2 × 10^-21

Details of the a lamellae examined after ageing at 400 K in Al-22 at.% Zn alloy

2 1

2 3 4 5

308 289 280 295 300

2.74 2.92 3.11 3.01 2.87

4.02 3.84 3.87 3.69 4.11

8.80 8.91 9.18 8.93 9.04

121 4.2 × 10^-21 3.5 × 10^-21 3.1 × 10^-21 3.5 × 10^-21 3.8 × 10^-21

3 1

2 3 4 5

154 167 143 148 159

1.84 1.75 1.71 1.68 1.62

3.54 3.76 3.91 3.62 3.82

7.10 7.13 7.19 6.90 6.97

57 0.7 × 10^-21 0.9 × 10^-21 0.7 × 10^-21 0.7 × 10^-21 0.9 × 10^-21

10 10^-18 10^-19 10^-20 10^-21

Cahn

Global approach

P-H Stationary GB

Al 4.33 Zn

9.39 Zn 16.77 Zn

49.4 Zn Local approach

16 20 24 28 32

10⁴/T [1/K]

10^-21 10^-22 10^-23 10^-24 10^-25

Al

Al-22 at.% Zn

10

ntent [at. %]

10 20 30 40 50

DD

EDX analysis after DP reaction at 400 and 450 K and then after DD reaction at 560 K and 570 K, for 3 single cells of different colonies

0.0 0.2 0.4 0.6 0.8 1.0

Spacing [arbitrary units]

5 6 7 8 9

Zn con

p = 8.14x10⁶ m-1 xi = 6.44 at.% Zn λα= 180 nm

DP

v

λ λλ λ_αααα

x´(y)

λ λ λ λ_αααα

0 y ¹

x*

DD-diffusion model

0 y ¹

K.N.Tu, D.B. Turnbull: Metall. Trans. A2 (1971) 2509 Assumption: p=z (small difference between DP and DD

temperatures)

P. Zięba, A. Pawłowski: Scripta Metall. 20 (1986) 1653 Separate description of p and z parameters (p≠z)

A,B,a,b = f(x_o, x_e, λ_α, p, x*, z)

xo

py z

p py b

z p zy a

B zy

A y

x +

−

−

− +

⋅ +

⋅

= sinh( ) cosh( ) cosh( ) sinh( )

)

´( λα λα 2 2 λα 2 2 λα

2 1 2

1

, 







=







=

b inst b

inst

D s z v D

s p v

δ δ

Growth velocity of the DD reaction

•In-situ observation in the TEM

•Directly from TEM micrographs (receding distance)

δ

16.7Zn

9.39Zn 4.33Zn

Pure Al

No. 2 DD

Stationary GB No. 4 DP No. 4 DD No. 3 DP

10^-19 10^-20 10^-21

14 18 22 26 30

10⁴/T [1/K]

s

δ

10^-22 10^-23 10^-24

Parameter Investigation

No. 1 No. 2 No. 3 No. 4 (This study)

v_DP Average by quantitative metallography

Average by quantitative metallography

Average by quantitative metallography

Directly from in situ observation

l_α Average by quantitative metallography

Directly from TEM micrographs

Directly from TEM mi- crographs

Directly from TEM micrographs

x_av X-ray analysis for 30% vo- X-ray analysis for 60% - -

x_av X-ray analysis for 30% vo- lume occupied by DP

X-ray analysis for 60%

volume occupied by DP

- -

p Indirectly from x_av - Directly from EDX ana-

lysis

Directly from EDX analysis

x* Average by X-ray analysis Directly from EDX analysis but as average for the whole sample

- Directly from EDX

analysis

v_DD Average by quantitative metallography

Average by quantitative metallography

- Directly from in situ observation

z Indirectly from x* and the formula

Indirectly from x* - Directly from EDX

analysis

α α α α

L

x_o

β β β

β(Ni₃Sn)

Sn [at.%]

Microstructure and EDX Analysis - Ni-4 at.% Sn

4 5 6 7

arameter C

C = 0.016_λα -0.19

0.0 0.2 0.4 0.6 0.8 1.0

y

1 2 3

Sn content [at.%] _λα^{= 320 nm}

150 200 250 300 350 400

λα [nm]

2 3 4

Pa

C=(pλλλλ_αααα)²

0.0 0.2 0.4 0.6 0.8 1.0

y

2 3 4 5 6 7

Sn content [at.%] b

EDX analysis for the 10 lamellae randomly chosen from 10 different colonies after ageing at 775, 825 and 875 K

3 /s ]

10^-19 10^-20 10^-21 10^-18

Frebel, Predel, Klisa DD

DP Stationary GBs

DD/DP:775 K DD/DP:800 K DD/DP:825 K DD/DP:840 K Global approach Local approach

P-H Cahn This study

This study

This study

950 850 750 K

P-H

10 11 12 13 14

10⁴/T [1/K]

sδD b [m3 10^-21

10^-22 10^-23 10^-24 10^-25

Cahn

L

Cu-In System

α

δ

Atomic percent In Cu x_o

Cu-4.5 At.% In

2.5 3.0

%]

0.5 µm

λλλ λ_αααα

0 40 80 120 160

Ort [nm]

1.0 1.5 2.0

In [At.%

Spacing [nm]

5 EDX line-scans in various colonies after ageing at 525, 550, 575, 600, 625, 650 K.

10^-21 10^-22 10^-17 10^-18 10^-19 10^-20

850 750 650 550 K

Global approach

Polycrystal Bicrystal Stationary GB

Local approach

Cahn Cahn P-H

P-H

10 12 14 16 18 20

10

/T [1/K]

10^-22

10^-24 10^-23

10^-25 10^-26

Co-13 At.% Al

RF

α α α α

4 6 8 10

[At.%]

0.2 µm

0 40 80 120

Ort [nm]

0 2

Al 4

Spacing [nm]

λ λ λ λ_αααα

Five EDX analysis taken in various colonies after DP reaction at 750, 800, 850, 900, 950, 1000 K.

Co-Al System

β (AlCo) _{1180 °C}

1495 °C L 1640 °C

0 10 20 30 40 50 60 70

Atomic percent aluminium

Co

α

Magnetic Transformation 1121 °C

422 °C

x_o

Al₅Co₂

εCo

[m3 /s]

Koncepcja lokalna

10^-19

10^-20 10^-21 10^-18

Co-13 Al

3 /s]

9 10 11 12 13 14 15

10⁴/T [1/K]

s

δ

10^-22 10^-23

10^-24 10^-25

sδD b [m3

Cu-Zn System

Chongmo-Hillert

Fe-Fe18.8 at.%Zn source

Chongmo, Hillert

DIGM:Fe-Fe18.8 at.%Zn source

Chuang et. al.: DP: Fe-13.5 at.%Zn

at.%Zn

Stationary GB Migrating GB

≈

≈

≈

≈

B∗

A

Db

s

δ

B∗

AB

D

s δ

B∗

AB

D

s δ

∗

D

s δ D

s δ

B∗

AB

Db

s

δ

A∗

AB/

≈

≈

A∗

AB

D

s δ

B∗

A

D

s δ

A∗

AB

Db

s

δ

B∗

A

D

s δ

A and B- solvent and solute atom, respectively

D^A/B*, D^AB/B* and D^AB/A*- tracer diffusion coefficients of B* in A, B in the alloy A-B, and A* in the alloy A-B^b

b b

Conclusions

Technique of analytical electron microscopy was shown as a valuable tool in characterisation of diffusion process along migrating grain boundaries of discontinuous precipitates

With careful assessment of experimental conditions, it seems likely that quantitative microanalyses of relatively high quality can be performed with an good spatial resolution approaching a few nanometers. This allows to determine the solute concentration profiles across the α lamellae (DP reaction) or left behind receding reaction front of DD and to compare them with the predictions of relevant theories

The use of the local concept of the DP reaction for Cahn´s model diminishes existing discrepancies in the diffusivity values in comparison with Petermann-Hornbogen model

Consequently, the reaction is no longer considered as mesoscopic phenomenon averaged over the whole volume of the sample but rather local event occurring in single cells

It is believed that the diffusivity values of the moving reaction front of the discontinuous precipitation and dissolution can be a source of reliable information about the diffusion rate, especially in systems and/or at temperatures where the radio- tracer data are not available

The diffusion along migrating and stationary GBs in Al-Zn, Cu- In, Ni-Sn and Ni-In systems occurs equaly fast